Absolute Value

Definition

Absolute value is a mathematical operation that describes the distance of a number from zero on the number line, regardless of whether the number is positive or negative. It represents the magnitude or size of a quantity without regard to its sign or direction.

5 Must Know Facts For Your Next Test

1. The absolute value of a number is always a non-negative real number, regardless of the original sign of the number.
2. Absolute value can be used to represent the distance between two points on the number line or in the coordinate plane.
3. Absolute value is denoted by vertical bars surrounding the number, like |x| or |5|.
4. Absolute value is an important concept in solving linear and absolute value equations and inequalities.
5. Absolute value functions have a distinctive V-shaped graph that reflects the distance of the input from the origin.

Review Questions

• Explain how the absolute value of a complex number is calculated and its significance in the context of complex numbers.
  • The absolute value of a complex number $z = a + bi$ is calculated as $|z| = \sqrt{a^2 + b^2}$. This represents the magnitude or modulus of the complex number, which is the distance from the origin to the point represented by the complex number in the complex plane. The absolute value of a complex number is always a non-negative real number and is an important property when performing operations with complex numbers, such as division and polar form representation.
• Describe how absolute value can be used to represent and solve linear and absolute value equations and inequalities involving complex numbers.
  • Absolute value can be used to represent and solve linear and absolute value equations and inequalities involving complex numbers. For example, the equation $|z - 3| = 2$ represents the set of all complex numbers $z$ that are 2 units away from the point 3 in the complex plane. To solve this equation, one would need to consider the two possible solutions: $z = 3 + 2i$ and $z = 3 - 2i$. Similarly, absolute value inequalities like $|z - 1| < 4$ can be used to describe the set of complex numbers that are within a certain distance of a given point in the complex plane, which is useful in various applications involving complex numbers.
• Analyze how the properties of absolute value, such as $|a| = |-a|$ and $|ab| = |a||b|$, can be used to simplify and manipulate complex number expressions.
  • The properties of absolute value, such as $|a| = |-a|$ and $|ab| = |a||b|$, can be very useful in simplifying and manipulating complex number expressions. The property $|a| = |-a|$ means that the absolute value of a complex number is the same regardless of its sign, which allows for simplification of expressions involving complex numbers. The property $|ab| = |a||b|$ allows for the absolute value of a product of complex numbers to be calculated by taking the product of the absolute values of the individual complex numbers. These properties can be leveraged to simplify complex number expressions, perform operations like division, and solve equations and inequalities involving complex numbers in the context of 3.1 Complex Numbers.

Related terms

Real Numbers: The set of all numbers that can be represented on the number line, including positive and negative integers, fractions, and irrational numbers.

Algebraic Expressions: Mathematical expressions involving variables and operations like addition, subtraction, multiplication, and division.

Inequalities: Mathematical statements that compare two quantities using symbols like greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤).